JEE MAIN - Mathematics (2022 - 28th June Evening Shift - No. 3)
The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives at least 4 and at most 7 candies, C3 receives at least 2 and at most 6 candies, is equal to :
205
615
510
430
Explanation
By multinomial theorem, no. of ways to distribute 30 identical candies among four children C1, C2 and C3, C4
= Coefficient of x30 in (x4 + x5 + .... + x7) (x2 + x3 + .... + x6) (1 + x + x2 ....)2
= Coefficient of x24 in $${{(1 - {x^4})} \over {1 - x}}{{(1 - {x^5})} \over {1 - x}}{{{{(1 - {x^{31}})}^2}} \over {{{(1 - x)}^2}}}$$
= Coefficient of x24 in $$(1 - {x^4} - {x^5} + {x^9}){(1 - x)^{ - 4}}$$
$$ = {}^{27}{C_{24}} - {}^{23}{C_{20}} - {}^{22}{C_{19}} + {}^{18}{C_{15}} = 430$$
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